Artificial Intelligence: Search Part 1: Uninformed graph search

Transcription

1 rtificial Intelligence: Search Part 1: Uninformed graph search Thomas Trappenberg January 8, 2009 ased on the slides provided by Russell and Norvig, hapter 3

2 Search outline Part 1: Uninformed search (tree search, graph search, etc) Part 2: Heuristic search (*, etc) Part 3 Optimization search algorithms (gradient decent, G, etc)

3 Selecting a state space Real world is absurdly complex state space must be abstracted for problem solving (bstract) state set of real states (bstract) action complex combination of real actions e.g., Halifax Hawaii represents a complex set of possible routes, detours, rest stops, anticipatory emotional, etc. (bstract) solution set of real paths that are solutions in the real world Each abstract action should be easier than the original problem!

4 Example: The 8-puzzle Start State Goal State states: integer locations of tiles (ignore intermediate positions) actions: move blank left, right, up, down (ignore unjamming etc.) goal test: = goal state (given) path cost: 1 per move [Note: optimal solution of n-puzzle family is NP-hard]

5 Example: Romania 71 Oradea Zerind 75 rad Timisoara Dobreta Neamt Iasi Sibiu Fagaras 80 Vaslui Rimnicu Vilcea 142 Lugoj Pitesti Mehadia Urziceni Hirsova 138 ucharest raiova Giurgiu Eforie

6 Tree search algorithms asic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) function TREE-SERH( problem, strategy) returns a solution, or failure initialize the search tree using the initial state of problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add the resulting nodes to the search tree end

7 Tree search example rad Sibiu Timisoara Zerind rad Fagaras Oradea Rimnicu Vilcea rad Lugoj rad Oradea

8 Tree search example rad Sibiu Timisoara Zerind rad Fagaras Oradea Rimnicu Vilcea rad Lugoj rad Oradea

9 Tree search example rad Sibiu Timisoara Zerind rad Fagaras Oradea Rimnicu Vilcea rad Lugoj rad Oradea

10 Implementation: states vs. nodes state is a (representation of) a physical configuration node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(x) States do not have parents, children, depth, or path cost! parent, action State state Node depth = 6 g = 6 The EXPND function creates new nodes, filling in the various fields and using the SUESSORFN of the problem to create the corresponding states.

11 Search strategies strategy is defined by picking the order of node expansion Strategies are evaluated along the following dimensions: completeness does it always find a solution if one exists? time complexity number of nodes generated/expanded space complexity maximum number of nodes in memory optimality does it always find a least-cost solution? Time and space complexity are measured in terms of b maximum branching factor of the search tree d depth of the least-cost solution m maximum depth of the state space (may be )

12 Uninformed search strategies Uninformed strategies use only the information available in the problem definition readth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search

13 readth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

14 readth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

15 readth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

16 readth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

17 Properties of breadth-first search omplete Yes (if b is finite) Time 1 + b + b 2 + b b d + b(b d 1) = O(b d+1 ), i.e., exp. in d Space O(b d+1 ) (keeps every node in memory) Optimal Yes (if cost = 1 per step); not optimal in general Space is the big problem; can easily generate nodes at 100M/sec so 24hrs = 8640G.

18 Uniform-cost search Expand least-cost unexpanded node Implementation: fringe = queue ordered by path cost, lowest first Equivalent to breadth-first if step costs all equal omplete Yes, if step cost ɛ Time # of nodes with g cost of optimal solution, O(b /ɛ ) where is the cost of the optimal solution Space # of nodes with g cost of optimal solution, O(b /ɛ ) Optimal Yes nodes expanded in increasing order of g(n)

19 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

20 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

21 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

22 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

23 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

24 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

25 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

26 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

27 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

28 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

29 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

30 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

31 Properties of depth-first search omplete No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space O(bm), i.e., linear space! Optimal No

32 Depth-limited search = depth-first search with depth limit l, i.e., nodes at depth l have no successors Recursive implementation: function DEPTH-LIMITED-SERH( problem, limit) returns soln/fail/cutoff REURSIVE-DLS(MKE-NODE(INITIL-STTE[problem]), problem, limit) function REURSIVE-DLS(node, problem, limit) returns soln/fail/cutoff cutoff-occurred? false if GOL-TEST(problem, STTE[node]) then return node else if DEPTH[node] = limit then return cutoff else for each successor in EXPND(node, problem) do result REURSIVE-DLS(successor, problem, limit) if result = cutoff then cutoff-occurred? true else if result failure then return result if cutoff-occurred? then return cutoff else return failure

33 Iterative deepening search function ITERTIVE-DEEPENING-SERH( problem) returns a solution inputs: problem, a problem for depth 0 to do result DEPTH-LIMITED-SERH( problem, depth) if result cutoff then return result end

34 Iterative deepening search l = 0 Limit = 0

35 Iterative deepening search l = 1 Limit = 1

36 Iterative deepening search l = 2 Limit = 2

37 Iterative deepening search l = 3 Limit = 3

38 Properties of iterative deepening search omplete Yes Time (d + 1)b 0 + db 1 + (d 1)b b d = O(b d ) Space O(bd) Optimal Yes, if step cost = 1 an be modified to explore uniform-cost tree Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(IDS) = , , , 000 = 123, 450 N(FS) = , , , , 990 = 1, 111, 1 IDS does better because other nodes at depth d are not expanded FS can be modified to apply goal test when a node is generated

39 Summary of algorithms riterion readth- Uniform- Depth- Depth- Iterative First ost First Limited Deepening omplete? Yes Yes No Yes, if l d Yes Time b d+1 b /ɛ b m b l b d Space b d+1 b /ɛ bm bl bd Optimal? Yes Yes No No Yes

40 Repeated states Failure to detect repeated states can turn a linear problem into an exponential one! D

41 Graph search function GRPH-SERH( problem, fringe) returns a solution, or failure closed an empty set fringe INSERT(MKE-NODE(INITIL-STTE[problem]), fringe) loop do if fringe is empty then return failure node REMOVE-FRONT(fringe) if GOL-TEST(problem, STTE[node]) then return node if STTE[node] is not in closed then add STTE[node] to closed fringe INSERTLL(EXPND(node, problem), fringe) end

42 More search algorithms bi-directional search,... see traversal

43 Summary Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored Variety of uninformed search strategies Iterative deepening search uses only linear space and not much more time than other uninformed algorithms Graph search can be exponentially more efficient than tree search